Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group 𝖧1

In this paper we prove that isοperimetric sets in three-dimensiοnal hοmogeneous spaces diffeοmorphic to ℝ3 are tοpological balls. Due to the work in [MMPR13], this settles the Uniqueness of Isοperimetric Dοmains Cοnjecture, concerning congruence of such sets. We also prove that in three-dimensiοnal homοgeneous spheres isοpermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensiοnal Heιsenberg grοup 𝖧1, characterizing the isοperimetric sets and constants for a family of Riemannιan adapted metrics. Using Γ-cοnvergence of the perimeter functiοnals, we also settle an isoperimetric conjecture in 𝖧1 posed by P. Paηsu.

Saturday, January 17, 2015

one of the most frustrating tasks is writing the problem set for the first homework assignment:

there's usually very little content from which to ask any interesting [1] problems;

for the student who lack self-awareness, it's best to have a few hard, interesting problems early on, if only so that they are not deluded into thinking that this will be an easy class [2] ..!

the problems shouldn't be impossible, either, otherwise done students will panic, expect everything to be hard, and won't be able to do as well as they could otherwise!

[1] from experience, if i think a problem is interesting then often it will be too hard for the students ..

[2] the trouble with teaching maths is that we always teach topics that we know with great certainty .. which means, without experience, it is very difficult to tell what is easy for the students and what is not. for one thing, i'm recently becoming aware that students never know linear algebra as well as i'd want them to know it..

Saturday, January 10, 2015

so i just wrote my first lecture of 2015 .. and it reads like algebra, if only because it's supposed to read that way.

it's a first lecture in an undergraduate course on complex analysis. i've never taught it before.

something worries me; i wonder if i'm the right guy to teach it.

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i'm not worried about screwing it up, not like last fall's numerical analysis ..

.. more on that, later ..

.. but now i have to come to terms with the subject. you see, i've deemed the subject unnecessary and consistently tried to avoid all aspects of complex analysis in my mathematical life.

yes, i know; it's the mathematical version of being a bigot or committing heresy!

perhaps it's just my first exposure to the subject [1]. perhaps it's the nature of the research problems i've chosen to study over the years. perhaps i generally take a pessimistic viewpoint in maths and a statement of the form ..

.. "every differentiable function at a point is infinitely differentiable at that same point and can be represented as a power series" ..

.. just sounds too good to be true, that there must have been some mistake. ye gods: what kind of dark, forbidden magic have we wrought?

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now, of all things, i have to teach it.

with luck, my students won't inherit my own prejudices on the matter. i'll even attempt to sell them on its good points, if only for the sake of being a responsible mathematical role model.

on a related note, my first lecture can be summed up as such:

if you take the usual coordinate page, view it as a 2-dimensional linear subspace of R^4, rotate it appropriately, and project it down to its original domain, then you can make sense of square roots of negative numbers, provided that you treat those image vectors as 2x2 matrices.

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notes:

[1] it's fair to say that my first course in it was uninspiring and largely i skipped all of the lectures except the exam periods. how i did "well" in that course is beyond me ..

.. but to be fair, i went to my favorite coffeehouse three times a week, armed with the textbook, ordered a large coffee, and would work through the problem sets from scratch. my barista friend would see me there regularly; she once asked me if i really liked that book or class, because i was reading it so much, to which i guffawed.

Wednesday, July 30, 2014

.. except that something has to change; this shouldn't happen in a country that calls itself a democracy.

Urban teachers have a kind of underground economy, Cohen explained. Some teachers hustle and negotiate to get books and paper and desks for their students. They spend their spare time running campaigns on fundraising sites like DonοrsChoose.org, and they keep an eye out for any materials they can nab from other schools. Philadelphia teachers spend an average of $300 to $\$$1,000 of their own money each year to supplement their $100 annual budget for classroom supplies, according to a Philadelphia Federation of Teachers survey.

Wednesday, July 23, 2014

this article is about how programming, despite the call to arms about learning how to code, is a low-status job.

when i read this post, though, it funded more like the plight of teachers:

.."that we allow “passion” to be used against us. When we like our work, we let it be known. We work extremely hard. That has two negative side effects. The first is that we don’t like our work and put in a half-assed effort like everyone else, it shows. Executives generally have the political aplomb not to show whether they enjoy what they’re doing, except to people they trust with that bit of information. Programmers, on the other hand, make it too obvious how they feel about their work. This means the happy ones don’t get the raises and promotions they deserve (because they’re working so hard) because management sees no need to reward them, and that the unhappy ones stand out to aggressive management as potential “performance issues”. The second is that we allow this “passion” to be used against us. Not to be passionate is almost a crime .."

Tuesday, July 22, 2014

So extreme are the admission standards now that kids who manage to get into elite colleges have, by definition, never experienced anything but success. The prospect of not being successful terrifies them, disorients them. The cost of falling short, even temporarily, becomes not merely practical, but existential. The result is a violent aversion to risk. You have no margin for error, so you avoid the possibility that you will ever make an error. Once, a student at Pomona told me that she’d love to have a chance to think about the things she’s studying, only she doesn’t have the time. I asked her if she had ever considered not trying to get an A in every class. She looked at me as if I had made an indecent suggestion.

like any news article on education, one should take this report with a reasonable amount of skepticism ..

.. but being a university educator myself, there's some truth in it. generally my students are uncomfortable when i ask them problems in the exam that don't match up with their textbook problems (even though they are usually combinations of the same problems). the risk of a new obstacle, of not having seen something on which they will be evaluated .. it seems to really affect them.

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for instance, last semester i think i spooked most of my linear algebra class with one geometry problem on each exam. [1] at some point several students asked for practice geometry problems.

"everyone's worried about the geometry problem," one of them admitted. i tried to point out that it was only one of at most five problems and that i generally curve the scores ..

.. but (s)he didn't seem convinced.

[1] e.g. "Determine, if it exists, an equation for the sphere passing through the following four points." (i even reminded them what the equation of a 2-sphere in 3-space was!)

in case you were wondering ..

yes, the names of well-known people and theorems are obscured. as for why, too many irrelevant Google searches find their way here.
for example, search: "mαth jοbs wιkι" but without the funny greek symbols. as of a year or two ago, this blog came up in the first few hits.